Primes and Composites

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Transcript of Primes and Composites

By: Audrey Marshall and Erica Buck Primes and Composites Essential Understandings: Natural Numbers: Multiplication Basic Facts (Mulitples) Important Prerequisite Understandings and Skills The Sieve of Eratosthenes Models and Representations Common Student Errors and/or Misconceptions Connections Key Language and Symbols Helpful Teacher Resources How? Why? Finding Primes The Fundamental Theorem of Arithmetic Factorizations Divisibility Rules Number Sense Powers Prime Numbers Composite Numbers: Factors: Product: Even Number: Odd Number: http://www.sheppardsoftware.com/mathgames/numbers/fruit_shoot_prime.htm A number that is multiplied by another number to find a product. Definition: Characteristics: The process of taking a number apart into factors. When the factors are multiplied together their product is the number you started with. In the first 100 numbers, 25 are prime. is the only even prime number is the smallest prime numberComposite numbers can be written as a product of only prime numbers (The Fundemental Theorm of Arithmetic) Examples Definition Emirp Numbers: Prime numbers that result in a different prime when its digits are reverse. 220 2 110 2 55 5 11 144 12 12 4 3 3 4 2 2 2 2 Common Divisibility Rules 2: A number is divisible by 2 if and only if it's ones digit is 0, 2, 4, 6, or 8. 3: A number is divisible by 3 if and only if the sum of it's digits is divisible by 3. 4: A number is divisible by 4 if and only if the number represented by its last two digits is divisible by 4. 5: A number is divisible by 5 if and only if its ones digit is divisible by 5; that is, if and only if the ones digit is 0 or 5. 6: A number is divisible by 6 if and only if it is divisible by both 2 and 3. 8: A number is divisible by 8 if an only if the number represented by its last three digits is divisible by 8. 9: A number is divisible by 9 if and only if the sum of its digits is divisible by 9. 10: A number is divisible by 10 if and only if its ones digit is divisible by 10; that is, if and only if the ones digit is 0. Why You need to know divisibility rules so that when you are figuring out factors and making prime factorizations you will know what numbers divide into each other. Examples 324 = 3+2+4=9/3=3 3 Divisibility test: 8 Divisibility test: 1324=1000 + 328 328/8= 41 Why You need to know factorizations to understand primes so that you know that primes only have two factors and composites have more than two. Multiplication chart from 1-12, showing all the multiples to 12 for each number. Why Basic multiplication facts will help you understand primes and composites because if you know the multiples of each number then you will know if they are composite or not because there will be more than two factors. Definition How many times a number is multiplied by itself. Description of Number Sense Definition: A natural number with exactly two factors; one and itself Examples: In The First 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 67, 71, 73, 79, 83, 89, 97 Some Larger Primes: 827, 353, 7919 (the 1,000Th prime) Non examples: All composite numbers. All even numbers (excluding 2) because it is special and neither prime nor composite 4, 25, 100, 9, etc. Examples: 13 and 31 (Prime spelled Backwards) 17 and 71 37 and 73 79 and 97, 107 and 701 157 and 751 Twin Primes: A prime number that differs from another prime number by two. Examples: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43) 1 2 3 4 5 Definition: Any natural number with more than two factors Characteristics: is the smallest composite numberThey can be written as a product of only primes (factor trees) to get prime factorizationsCan be even or odd numbers Examples: All even numbers, excluding (4, 6, 10, 20, 98) Some odd numbers(9, 93, 33, 45, 15, 49) Non Examples: Any number that is prime: 3, 2, 17, 11, etc. is neither prime nor composite Multiples Definition: The counting numbers (1, 2, 3, …) Characteristics: They are the numbers we use to count with Positive numbers 0 (zero) is not included Definition: A number that can be divided or separated evenly into two equal groups. Characteristics: Always ends with a 0, 2, 4, 6, or 8 Always divisible by 2 1 more or 1 less that and odd number Even # = 2n Examples: 2, 8, 16, 24, 68, 100 8 8 4 4 2 2 1 1 Non Examples: Numbers that cannot be divided by 2 (odd numbers) 1, 3, 5, 7, 9, 11, ... Definition: A number that cannot be divided equally into two equal groups. Characteristics: Always end in a: 1, 3, 5, 7, or 9 Always 1 left over 2n+1 Examples: 7, 15, 33, 21, 97 15 7 8 = 3 5 16 8 6 Non Examples: All even numbers 2, 14, 26, 38, 70 4 3 , 3 With the divisibility test for 8, you look at the last three digits and see if it goes into 8. Why it matters? For three, you take the sum of the digits of the number and see if they go into three. Odd numbers are important when it comes to prime and composite numbers, because odd numbers are not divisible by 2 which lowers the probability of having more than two factors. All prime numbers are odd numbers, except for the numbers & 1 is neither prime nor composite and 2 is the only even prime number. . These are factor trees showing the factors of both 220 and 144, which are both composite numbers but they have prime factors within them. Why it matters? Even numbers are important in regard to prime and composite numbers because all even numbers are composite numbers because they have more than two factors. 2 is the exception to this rule because its only factors are 1 and itself. This makes it the only even prime number. 4 X X has 3 factors. 4 is a composite number. Why it matters? It is important to know what natural numbers are because prime numbers only apply to natural numbers. If it is not clear that primes are only natural counting numbers, then it can cause confusion and mistakes with students. This center diagonal shows the square numbers, these are a product of a number times itself. The rows highlighted in yellow show the identity property. This means that any number times one equals itself. Definition: The product is what the answer is called when two numbers are multiplied together. The product is also a multiple of the the numbers. Examples: X = (8 is the product) The rows highlighted in orange show the doubles 2's structure. These show you the multiples of 2. is a multiple of both 4 and 2 Definition: A multiple of a number can divide a number evenly without any remainders. Multiples of a number are created by multiplying a number by the other numbers. The rows with fours at the top of them show the doubles 4's structure. These show you the multiples of 4. infinite number of primes Examples The 5 is multiplied by itself three times. The 2 is multiplied by itself four times. why The power of a prime number will tell you how many factors the number has by knowing that whatever power the number has is one more than the the number of factors the number has. Why It Matters? It is important to know what the product of two numbers, because the product is also a factor of the two numbers. This is important to know, because the number of factors a number has determines whether or not a number is prime or composite. Examples: 3 33 11 4 x 3 = 12 An understanding of numbers, their magnitude, relationships, and how they are affected by operations. The ability to work outside the traditionally taught algorithms. 12 is a multiple of 4. 12 is also a multiple of 3. Why it Matters? Multiples are important in regards to prime and composite numbers because a number that is a multiple of another number is created with more than 2 factors, making it a composite number. Example: X = why 4 is a factor of 12. 5 is a factor of 12. Understanding Number Sense is important for understanding primes and composites because you need to have a firm foundation of numbers. With this foundation you will be able to understand numbers and the many ways that they work. Why it matters? It is important to know factors because the number of factors a number has determines whether a number is prime or composite. The Sieve of Eratothenes is an algorithm for finding all prime numbers up to any given limit. It marks all the multiples of each prime as composite, starting with the multiples of 2. The multiples of a given prime are generated starting from that prime, as a sequence of numbers with the same difference, equal to that prime, between consecutive numbers. To find all the prime numbers less than or equal to a given integer n. Create a list of consecutive integers from 2 to nInitially, let p equal 2, the first prime number. Starting from p, count up in increments of p and mark each of these numbers greater than p itself in the list. These will be multiples of p: 2p, 3p, 4p, etc. note that some of them have already been marked. If there was no such number, stop. Otherwise, let p now equal this number and repeat from step 3. When the algorithm terminates, all the numbers in the list that are not marked are prime. This is a helpful algorithm to know to understand primes and composites. With a chart you can do the algorithm yourself and see which numbers are primes and composites. You will be able to understand that primes only have factors of 1 and themselves and composites have a multiple number of factors. Factor Trees Here is a link that will allow students to do factor trees virtually and understand what they are. http://www.mathplayground.com/factortrees.html Why it Helps This manipulative will help students understand many factors of numbers. It will also help them understand prime factorizations of numbers. A prime number is a natural number that has exactly two different factors, and the number itself (see prime in key language for more information) A composite number is a natural number that has more than two different factors. (see composite in key language for more information) The number is neither prime nor composite because it has only one factor, itself. It is considered a special number. (see one in common student error/misunderstandings for more information) Prime or composite numbers can be represented by rectangular arrays on grid paper. A prime number can be represented by only one rectangular array (e.g., 7 can be represented by a 7 ´ 1 and a 1 x 7). A composite number can always be represented by more than two rectangular arrays (e.g., 9 can be represented by a 9 ´ 1, a 1 x 9, or a 3 ´ 3). Divisibility rules aid in identifying prime and composite numbers. (see divisibility rules in prerequisites for more information) Definition: The Fundamental Theorem of Arithmetic states: Any integer greater that is either a prime number, or can be written as a unique product of prime numbers. Factor Trees (see factor trees in models for more information) Arrays Prime factorization concepts can be created using factor trees. 70 7 X 10 Prime Number 2 X 5 70 = 7 x 2 x 5 Books Picture example Students may be confused as to whether the number is prime or composite. is neither prime or composite. It is special in that it is a factor of every number. It is the only number with one factor, itself. Games Websites and Videos Students may also get confused as to whether or not the number is prime or composite. Fruit Shoot: Primes and Composites is a prime number and is the only even prime number. Shoot the fruit while the target switches from composite to primes. Play with prime numbers to 20, 50, or 99. Game also has a relaxed mode and a timed mode for more intensity! King Kong's Prime Numbers A third number students might be confused whether it is prime or composite is In this whack-a-mole like game, the player "whacks" King Kong when he is holding a prime number in order to get points. But beware, if you click on a composite number, you will loose a point. Use this interactive web page when teaching about the Sieve of Eratosthenese. <a href="http://www.softschools.com/math/prime_numbers/prime_numbers_up_to_100/">Prime Numbers Up To 100</a> Prime Numbers Up To 100 Why Prime Landing is neither prime or composite because it is not considered a counting number. For a number to be prime it has to be a counting number. Land your rocket ship safely by clicking on the yes or no button on whether the number is prime or composite. Why Why A fourth number that students may confuse whether it is prime or composite is Prime Number Rap Use this as a mnemonic device for memorizing prime numbers. Why Prime and Composite Numbers is composite because it's factors are 1, 5, and 25. Students may think it is prime because 5 is prime and 25 ends in 5. This video gives a nice demonstration of what prime and composite numbers are. by Richard Evan Schwartz You Can Count on Monsters A great visual for students that explores factoring as well as prime and composite numbers. The author creates monsters for each of the numbers. Locate items in the grocery that are packaged in a way representing composite numbers, and if possible, prime numbers. Ask students to explain why grocery items, such as eggs, are packed in sets of 12, not 11 or 13. Encourage them to consider the arrangement of the possible arrays. The Grapes Of Math Real World Application This book is filled with math riddles that can be solved by teaching students math theory concepts such a prime and composite numbers. Finding a Numbers Factors Because of prime factorization, you can continue onto using it to find how many numbers a factor has. Example: 16 Number Prime Factorization List of Factors Number of Factors 2 4 1, 2, 4, 8, 16 5 The pattern is to take the number in the power, in this case 4, and add 1. Power + 1 = # of factors. Why it matters? This is a great tool in math, especially when solving a problem that asks how many factors a number has, or you can work backwards and answer a question that gives you the amount of factors and know that the prime factorization of the number is a prime with a power of one less that what you started with. Cryptography What it is. Writing and solving codes for security, credit cards, and other types of technically related sources. Why Primes? A secret Language It is easier to make large prime numbers than it is to factor large numbers into primes. This makes it easier to make codes and secret messages. Example 2436895187740522149300895060339985963357828798391070516253607140448055114932771201027350325, 32391566613318777717463374330766574149515851358738762166744284515065903121845841724822236676 Encrypted Message The number N is the product of two primes, just as 39 is the product of the primes 3 and 13. If you could discover the primes, you could decrypt the message. But this is not so easy to do, and this is one of the reasons why encrypting works. e = 5, N = 519208104502047440191322024032 461128846299254256408973265508 51544998255968235697331455544257 Encrypted using the key: (see fundamental theorem of arithmetic in understandings for more information)